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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 15
Chapter 6, Problem 15

Solve each system in Exercises 5–18. {x+y=−4y−z=12x+y+3z=−21\(\begin{cases}\)x + y = -4 \(\y\) - z = 1 \\2x + y + 3z = -21\(\end{cases}\)

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Write down the system of equations clearly: \[x + y = -4\] \[y - z = 1\] \[2x + y + 3z = -21\]
From the first equation, express one variable in terms of the other. For example, solve for \[y\]: \[y = -4 - x\]
Substitute the expression for \[y\] into the second equation to relate \[x\] and \[z\]: \[(-4 - x) - z = 1\]
Simplify the equation from step 3 to express \[z\] in terms of \[x\]: \[z = -5 - x\]
Substitute the expressions for \[y\] and \[z\] from steps 2 and 4 into the third equation: \[2x + (-4 - x) + 3(-5 - x) = -21\], then simplify and solve for \[x\].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Such systems can be solved using substitution, elimination, or matrix methods.
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Substitution and Elimination Methods

Substitution involves solving one equation for a variable and substituting that expression into other equations. Elimination involves adding or subtracting equations to eliminate a variable, simplifying the system. Both methods help reduce the system to fewer variables for easier solving.
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Three-Variable Systems

When a system has three variables, it typically requires solving three equations simultaneously. Understanding how to manipulate and combine equations to isolate variables is essential. Solutions can be unique, infinite, or nonexistent depending on the system's consistency.
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Related Practice
Textbook Question

Solve each system in Exercises 5–18. {3(2x+y)+5z=−12(x−3y+4z)=−94(1+x)=−3(z−3y)\(\begin{cases}\)3(2x + y) + 5z = -1 \\2(x - 3y + 4z) = -9 \\4(1 + x) = -3(z - 3y)\(\end{cases}\)

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Textbook Question

Write the partial fraction decomposition of each rational expression. x/(x2 +2x -3)

Textbook Question

In Exercises 16–24, write the partial fraction decomposition of each rational expression. x/(x - 3)(x + 2)

Textbook Question

In Exercises 1–18, solve each system by the substitution method. {x+y=1x2+xy−y2=−5\(\begin{cases}\)x + y = 1 \(\x\)^2 + xy - y^2 = -5\(\end{cases}\)

Textbook Question

Write the partial fraction decomposition of each rational expression. 4/(2x2 -5x -3)

Textbook Question

In Exercises 1–26, graph each inequality. x2+y2>25

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